Oh cool I've never seen that pic. I was meaning to get that book but forgot about it. Anyway I remember reading Lou and Sterling got 12-string guitars for the third album so I would guess a song off that? Maybe like I'm Set Free or something.

It's a Fender Elecrtic XII, they were introduced in '65, but most production was '66-69, then they were dropped. It was not a popular gtr, but it had versatile pickup switching options that were not available on other gtrs if the time and the construction meant they had a heavier sound than other more typical electric 12 strings. I imagine these aspects might have appealed to Lou...

12 string played by Lou at outdoor show in daylight "live in Texas, 1969"
parts of 12 string visible in pictures at page 52/53 in VU 45th Anniversary Edition

notes:
*in Yule's photobook "69 on the road" [which I don't own, nor have ever seen] are possibly more pictures with that 12 string
*have been wondering earlier, where that outdoor daylight show might have been, but never came to ask here
*just checked Olivier's site, and noticed a for me new entry of "1969.Oct.15 on the Bank of White Rock Lake, Dallas, TX", could that be the location?
[other entries for Texas End of Cole in Dallas and Vulcan Gas in Austin, both in 1969 October, but no mention of any outdoor performace]

David Fricke mentions on pg. 11 of the Third Album 45th anniv. reissue book:

Morrison noted that he and Reed played Fender electric 12-string guitars on most of The Velvet Underground, giving the arrangements an airy luxury that heightened details like the rubbery trem-bar shivers in "Some Kinda Love" and Tucker's mounting-heartbeat rhythm in "I'm Set Free."